(0) Obligation:

Clauses:

sum(X, 0, X).
sum(X, s(Y), s(Z)) :- sum(X, Y, Z).

Query: sum(a,g,a)

(1) PrologToPrologProblemTransformerProof (SOUND transformation)

Built Prolog problem from termination graph ICLP10.

(2) Obligation:

Clauses:

sumA(T5, 0, T5).
sumA(T19, s(0), s(T19)).
sumA(T29, s(s(T27)), s(s(T30))) :- sumA(T29, T27, T30).

Query: sumA(a,g,a)

(3) PrologToPiTRSProof (SOUND transformation)

We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
sumA_in: (f,b,f)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

sumA_in_aga(T5, 0, T5) → sumA_out_aga(T5, 0, T5)
sumA_in_aga(T19, s(0), s(T19)) → sumA_out_aga(T19, s(0), s(T19))
sumA_in_aga(T29, s(s(T27)), s(s(T30))) → U1_aga(T29, T27, T30, sumA_in_aga(T29, T27, T30))
U1_aga(T29, T27, T30, sumA_out_aga(T29, T27, T30)) → sumA_out_aga(T29, s(s(T27)), s(s(T30)))

The argument filtering Pi contains the following mapping:
sumA_in_aga(x1, x2, x3)  =  sumA_in_aga(x2)
0  =  0
sumA_out_aga(x1, x2, x3)  =  sumA_out_aga
s(x1)  =  s(x1)
U1_aga(x1, x2, x3, x4)  =  U1_aga(x4)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

Pi-finite rewrite system:
The TRS R consists of the following rules:

sumA_in_aga(T5, 0, T5) → sumA_out_aga(T5, 0, T5)
sumA_in_aga(T19, s(0), s(T19)) → sumA_out_aga(T19, s(0), s(T19))
sumA_in_aga(T29, s(s(T27)), s(s(T30))) → U1_aga(T29, T27, T30, sumA_in_aga(T29, T27, T30))
U1_aga(T29, T27, T30, sumA_out_aga(T29, T27, T30)) → sumA_out_aga(T29, s(s(T27)), s(s(T30)))

The argument filtering Pi contains the following mapping:
sumA_in_aga(x1, x2, x3)  =  sumA_in_aga(x2)
0  =  0
sumA_out_aga(x1, x2, x3)  =  sumA_out_aga
s(x1)  =  s(x1)
U1_aga(x1, x2, x3, x4)  =  U1_aga(x4)

(5) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

SUMA_IN_AGA(T29, s(s(T27)), s(s(T30))) → U1_AGA(T29, T27, T30, sumA_in_aga(T29, T27, T30))
SUMA_IN_AGA(T29, s(s(T27)), s(s(T30))) → SUMA_IN_AGA(T29, T27, T30)

The TRS R consists of the following rules:

sumA_in_aga(T5, 0, T5) → sumA_out_aga(T5, 0, T5)
sumA_in_aga(T19, s(0), s(T19)) → sumA_out_aga(T19, s(0), s(T19))
sumA_in_aga(T29, s(s(T27)), s(s(T30))) → U1_aga(T29, T27, T30, sumA_in_aga(T29, T27, T30))
U1_aga(T29, T27, T30, sumA_out_aga(T29, T27, T30)) → sumA_out_aga(T29, s(s(T27)), s(s(T30)))

The argument filtering Pi contains the following mapping:
sumA_in_aga(x1, x2, x3)  =  sumA_in_aga(x2)
0  =  0
sumA_out_aga(x1, x2, x3)  =  sumA_out_aga
s(x1)  =  s(x1)
U1_aga(x1, x2, x3, x4)  =  U1_aga(x4)
SUMA_IN_AGA(x1, x2, x3)  =  SUMA_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4)  =  U1_AGA(x4)

We have to consider all (P,R,Pi)-chains

(6) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

SUMA_IN_AGA(T29, s(s(T27)), s(s(T30))) → U1_AGA(T29, T27, T30, sumA_in_aga(T29, T27, T30))
SUMA_IN_AGA(T29, s(s(T27)), s(s(T30))) → SUMA_IN_AGA(T29, T27, T30)

The TRS R consists of the following rules:

sumA_in_aga(T5, 0, T5) → sumA_out_aga(T5, 0, T5)
sumA_in_aga(T19, s(0), s(T19)) → sumA_out_aga(T19, s(0), s(T19))
sumA_in_aga(T29, s(s(T27)), s(s(T30))) → U1_aga(T29, T27, T30, sumA_in_aga(T29, T27, T30))
U1_aga(T29, T27, T30, sumA_out_aga(T29, T27, T30)) → sumA_out_aga(T29, s(s(T27)), s(s(T30)))

The argument filtering Pi contains the following mapping:
sumA_in_aga(x1, x2, x3)  =  sumA_in_aga(x2)
0  =  0
sumA_out_aga(x1, x2, x3)  =  sumA_out_aga
s(x1)  =  s(x1)
U1_aga(x1, x2, x3, x4)  =  U1_aga(x4)
SUMA_IN_AGA(x1, x2, x3)  =  SUMA_IN_AGA(x2)
U1_AGA(x1, x2, x3, x4)  =  U1_AGA(x4)

We have to consider all (P,R,Pi)-chains

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 1 less node.

(8) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

SUMA_IN_AGA(T29, s(s(T27)), s(s(T30))) → SUMA_IN_AGA(T29, T27, T30)

The TRS R consists of the following rules:

sumA_in_aga(T5, 0, T5) → sumA_out_aga(T5, 0, T5)
sumA_in_aga(T19, s(0), s(T19)) → sumA_out_aga(T19, s(0), s(T19))
sumA_in_aga(T29, s(s(T27)), s(s(T30))) → U1_aga(T29, T27, T30, sumA_in_aga(T29, T27, T30))
U1_aga(T29, T27, T30, sumA_out_aga(T29, T27, T30)) → sumA_out_aga(T29, s(s(T27)), s(s(T30)))

The argument filtering Pi contains the following mapping:
sumA_in_aga(x1, x2, x3)  =  sumA_in_aga(x2)
0  =  0
sumA_out_aga(x1, x2, x3)  =  sumA_out_aga
s(x1)  =  s(x1)
U1_aga(x1, x2, x3, x4)  =  U1_aga(x4)
SUMA_IN_AGA(x1, x2, x3)  =  SUMA_IN_AGA(x2)

We have to consider all (P,R,Pi)-chains

(9) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(10) Obligation:

Pi DP problem:
The TRS P consists of the following rules:

SUMA_IN_AGA(T29, s(s(T27)), s(s(T30))) → SUMA_IN_AGA(T29, T27, T30)

R is empty.
The argument filtering Pi contains the following mapping:
s(x1)  =  s(x1)
SUMA_IN_AGA(x1, x2, x3)  =  SUMA_IN_AGA(x2)

We have to consider all (P,R,Pi)-chains

(11) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(12) Obligation:

Q DP problem:
The TRS P consists of the following rules:

SUMA_IN_AGA(s(s(T27))) → SUMA_IN_AGA(T27)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.

(13) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

  • SUMA_IN_AGA(s(s(T27))) → SUMA_IN_AGA(T27)
    The graph contains the following edges 1 > 1

(14) YES